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Round about Theta. Part I Prehistory

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There is a huge amount of work on different kinds of theta functions, the theta correspondence, cohomology classes coming from special Schwartz classes via theta distribution, and much more. The aim of this text is to try to find joint construction principles while often leaving aside relevant but cumbersome details. The next steps after this prehistoric Part I will be directed to a description of the Howe operators introduced by Kudla and Millson and their special Schwartz forms and classes. This has as attractor the fact that the modular and automorphic forms arising naturally in context with these classes find very nice geometric interpretations of their Fourier coefficients and thus lead to an intriguing intertwining of elements of representation theory with algebraic and arithmetic geometry. The presentation here is in the spirit of my book on representations of linear groups. Though it may be seen as just another chapter, it has it has its own raison d’être and

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ture quelconque (s,t) , soit LcRn un r~seau sur lequel q(x) prend des valeurs enti~res et soit p(x) : R n ~ C une fonction avec les propri~t~s suivantes : *) La fonction f(x) = p(x) e -2~q(x) , ainsi que D(x)f(x) et R(x)f(x) pour toute d~rivation D(x) d'ordre ~ 2 et tout polynSme R(x) de degr~ ~ 2 sont d~finies et appartiennent L2(R n) n Ll(e n) (E-X)p(x) = ~ p(x) , pour un entier ~ , ou E est l'op~rateur d'Euler et ~ le laplacien associ~ ~ q(x) . Alors la s~rie th~ta
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